Need to put this in the form of an ellipse equation : $4x^2 + 4x + y^2 =0$

Question

It needs to be in the form: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a,b \in \mathbb{R}.$$

I have tried:

$$4x(x+1) + y^2 = 0$$ but it doesn't give me the $1$ and divides $y^2$ by an $x$.

I have also tried: $$4x^2+y^2 = 4x$$ $$4x + (y^2/4x) = 1$$ Which gives me the $1$ but still divides the $y^2$ by an $x$.

Thanks for any help.

Answer 1

Hint

Note that

$$4x^2+4x=(2x+1)^2-1.$$

Answer 2

We have $$ 4x^2+4x+y^2=0 \iff 4x^2+4x+1+y^2-1=0 \iff (2x+1)^2+y^2=1 $$ so: $$ 4\left(x+\frac{1}{2}\right)^2 +y^2=1 $$